Question

1. identify whether each table, graph, or equation is linear, exponential, or neither.

a.

x	f(x)
-2	3
-1	4.5
0	6
1	7.5
2	9

b.

x	f(x)
0	2
1	6
2	18
3	52
4	156

c.
graph of a parabola opening upwards with vertex at (0, -2)
d.
graph of a curve decreasing from left to right, passing through (0, 2) and (3, 0)
e. $f(x) = \frac{5}{3}(0.2)^x$
f. $f(x) = \frac{3}{2}x + 2$

Explanation:

Part a:

Step1: Check linear (constant Δf(x))

Calculate differences:
$4.5-3=1.5$, $6-4.5=1.5$, $7.5-6=1.5$, $9-7.5=1.5$

Step2: Confirm linearity

Constant first difference, so linear.

Part b:

Step1: Check linear (Δf(x))

Calculate differences:
$6-2=4$, $18-6=12$, $52-18=34$, $156-52=104$

Step2: Check exponential (ratio f(x))

Calculate ratios:
$\frac{6}{2}=3$, $\frac{18}{6}=3$, $\frac{52}{18}\approx2.89$, $\frac{156}{52}=3$

Step3: Classify

No constant diff/ratio, so neither.

Part c:

Step1: Analyze graph shape

Graph is a parabola (U-shaped, quadratic)

Step2: Classify

Not linear/exponential, so neither.

Part d:

Step1: Analyze graph shape

Graph is decreasing, curved, matches exponential decay

Step2: Classify

Follows exponential form, so exponential.

Part e:

Step1: Match to exponential form

Exponential form: $f(x)=ab^x$
Given: $f(x)=\frac{5}{3}(0.2)^x$, fits $a=\frac{5}{3}$, $b=0.2$

Step2: Classify

Matches exponential structure, so exponential.

Part f:

Step1: Match to linear form

Linear form: $f(x)=mx+b$
Given: $f(x)=\frac{3}{2}x+2$, fits $m=\frac{3}{2}$, $b=2$

Step2: Classify

Matches linear structure, so linear.

Answer:

a. Linear
b. Neither
c. Neither
d. Exponential
e. Exponential
f. Linear